Iterative refinement for constrained and weighted linear least squares

Gulliksson, Mårten

doi:10.1007/bf01955871

Cited by 19 publications

(21 citation statements)

References 5 publications

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“…Although their condition numbers can be cheaply estimated, their norms are different from the infinity-norm we are using. The refinement algorithms for weighted and linearly contrained least squares also exist [9,15].…”

Section: Related Workmentioning

confidence: 99%

Extra-Precise Iterative Refinement for Overdetermined Least Squares Problems

Demmel

Hida

Riedy

et al. 2009

ACM Trans. Math. Softw.

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We present the algorithm, error bounds, and numerical results for extra-precise iterative refinement applied to overdetermined linear least squares (LLS) problems. We apply our linear system refinement algorithm to Björck's augmented linear system formulation of an LLS problem. Our algorithm reduces the forward normwise and componentwise errors to O(ε) unless the system is too ill conditioned. In contrast to linear systems, we provide two separate error bounds for the solution x and the residual r. The refinement algorithm requires only limited use of extra precision and adds only O(mn) work to the O(mn 2 ) cost of QR factorization for problems of size m-by-n. The extra precision calculation is facilitated by the new extended-precision BLAS standard in a portable way, and the refinement algorithm will be included in a future release of LAPACK and can be extended to the other types of least squares problems. BackgroundThis article presents the algorithm, error bounds, and numerical results of the extra-precise iterative refinement for overdetermined least squares problem (LLS):where A is of size m-by-n, and m ≥ n.The xGELS routine currently in LAPACK solves this problem using QR factorization. There is no iterative refinement routine. We propose to add two routines: xGELS_X (expert driver) and xGELS_RFSX (iterative refinement). In most cases, users should not call xGELS_RFSX directly.Our first goal is to obtain accurate solutions for all systems up to a condition number threshold of O(1/ε) with asymptotically less work than finding the initial solution. To this end, we first *

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Section: Related Workmentioning

confidence: 99%

Extra-Precise Iterative Refinement for Overdetermined Least Squares Problems

Demmel

Hida

Riedy

et al. 2009

ACM Trans. Math. Softw.

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“…Soderkvist has proposed an algorithm based on constrained and weighted least squares which uses as a main tool the weighted QRD [6,7]. Earlier Paige considered the GLM as a Generalized Linear Least-Squares Problem (GLLSP) and employed the generalized QRD (GQRD) to solve it [26].…”

Section: Introductionmentioning

confidence: 99%

Parallel Givens Sequences for Solving the General Linear Model on a Erew Pram∗

Kontoghiorghes

2000

Parallel Algorithms and Applications

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Parallel Givens sequences for solving the General Linear Model (GLM) are developed and analyzed. The block updating GLM estimation problem is also considered. The solution of the GLM employs as a main computational device the Generalized QR Decomposition, where one of the two matrices is initially upper triangular. The proposed Givens sequences efficiently exploit the initial triangular structure of the matrix and special properties of the solution method. The complexity analysis of the sequences is based on a Exclusive Read-Exclusive Write (EREW) Parallel Random Access Machine (PRAM) model with limited parallelism.Furthermore, the number of operations performed by a Givens rotation is determined by the size of the vectors used in the rotation. With these assumptions one conclusion drawn is that a sequence which applies the smallest number of compound disjoint Givens rotations to solve the GLM estimation problem does not necessarily have the lowest computational complexity. The various Givens sequences and their computational complexity analyses will be useful when addressing the solution of other similar factorization problems.

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“…Actually, system (1 .2) defines the whole class of constrained and weighted least squares problem, see [10] and [11] . More precisely, we may assume that the first p diagonal elements in M are zero and the rest diagonal elements are nonzero .…”

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confidence: 99%

A preconditioner for constrained and weighted least squares problems with Toeplitz structure

Jin

1996

Bit Numer Math

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.We study methods for solving the constrained and weighted least squares problem minim a (b -Ax) T W (b -Ax) by the preconditioned conjugate gradient (PCG) method . k. It is well-known that this problem can be solved by solving an augmented linear 2 x 2 block linear system MA +Ax = b, AT A = 0, where M = W -1 . We will use the PCG method with circulant-like preconditioner for solving the system . We show that the spectrum of the preconditioned matrix is clustered around one . When the PCG method is applied to solve the system, we can expect a fast convergence rate .

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Iterative refinement for constrained and weighted linear least squares

Cited by 19 publications

References 5 publications

Extra-Precise Iterative Refinement for Overdetermined Least Squares Problems

Extra-Precise Iterative Refinement for Overdetermined Least Squares Problems

Parallel Givens Sequences for Solving the General Linear Model on a Erew Pram∗

A preconditioner for constrained and weighted least squares problems with Toeplitz structure

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